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Robust estimation of seismic coda shape

Mikko Nikkilä1, Valentin Polishchuk

1,2 and Dmitry Krasnoshchekov

3*

1Helsinki Institute for Information Technology, CS Dept, University of Helsinki, Finland

2Communications and Transport Systems, ITN, Linkoping University, Sweden

3Institute of Dynamics of Geospheres, Leninsky pr. 38 korp. 1, Moscow 119334 Russia

* To whom correspondence should be addressed, E-mail: krasnd@idg.chph.ras.ru

Summary

We present a new method for estimation of seismic coda shape. It falls into the same class of

methods as non-parametric shape reconstruction with the use of neural network techniques where

data are split into a training and validation datasets. We particularly pursue the well-known

problem of image reconstruction formulated in this case as shape isolation in the presence of a

broadly defined noise. This combined approach is enabled by the intrinsic feature of seismogram

which can be divided objectively into a pre-signal seismic noise with lack of the target shape, and

the remainder that contains scattered waveforms compounding the coda shape. In short, we

separately apply shape restoration procedure to pre-signal seismic noise and the event record,

which provides successful delineation of the coda shape in the form of a smooth almost non-

oscillating function of time.

The new algorithm uses a recently developed generalization of classical computational-geometry

tool of α-shape. The generalization essentially yields robust shape estimation by ignoring locally

a number of points treated as extreme values, noise or non-relevant data. Our algorithm is

conceptually simple and enables the desired or pre-determined level of shape detail, constrainable

by an arbitrary data fit criteria.

The proposed tool for coda shape delineation provides an alternative to moving averaging and/or

other smoothing techniques frequently used for this purpose.

The new algorithm is illustrated with an application to the problem of estimating the coda

duration after a local event. The obtained relation coefficient between coda duration and

epicentral distance is consistent with the earlier findings in the region of interest.

1 Introduction

Continuous wavetrains that follow direct P- and S-arrivals on broadband records of seismic

events are called coda and generally associated with random heterogeneities in the lithosphere. It

has long been known that seismic coda shape possesses several robust features; for example,

temporal shape of narrow-band filtered coda is independent of epicentral distance for local events

(Rautian & Khalturin 1978). Many properties of coda shape, like its decay gradient or duration,

proved to be helpful in geophysical studies by providing important information on seismic source

and propagating media properties (see e.g. (Sato & Fehler 1998) for a review). The shape of local

earthquake coda was first formalized by Aki (1969) as a superposition of incoherent seismic

waves coming from uniformly distributed discrete scatterers.

The imaginary shape of seismic coda is hard to recognize by eye on seismic records of

displacement or velocity. Taking envelope of seismic record makes coda shape more explicit –

the envelope is essentially a presentation of seismic record’s information in the other, more

amenable and visually accessible form. This operation of calculation of analytic envelope is a

transfer of seismic record to another form of presentation rather than processing because it does

not distort the original record. The resulting envelope more prominently displays the decay

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character of the seismic signal amplitude in time. To extract useful geophysical information,

various signal processing techniques are applied then to the envelopes.

Earlier studies predominantly used squared (or root-mean-squared) envelopes, while more recent

works began employing other instruments too. For example, Nakahara & Carcolé (2010) used the

analytic envelopes. They invoked parametric regression to give Maximum-Likelihood estimates

of coda parameters under the assumption that the generic form of the decay function and noise

distribution are known. One of the advantages provided by analytic envelopes is that they

preserve phase information of the raw data and can be used as a measure of the time variation of

the total energy (kinetic and potential) involved in the seismic response. At the same time,

despite their increased amenability, analytic envelopes are still strongly oscillating functions that

require smoothing in order to enable further uncomplicated processing. Without smoothing,

difficulties may arise, for example, when estimating seismic attenuation by fitting a model curve

to specific portions of the envelope. Indeed, the commonly used least-square fitting procedure is

known to heavily weigh extreme values, which makes smoothing essential.

Among a number of smoothing procedures, the simple moving averaging remains the prevailing

one. The method is easy to grasp and implement, yields fast results and has only one control

parameter – sliding window length (some examples of the envelopes smoothed in various time

windows 1 to 100 seconds long are given in Fig. S1). At the same time, in general there exists no

single criterion for selection of the window length, so in practice the selection is done to decrease

fluctuations below a certain level.

In this article we propose not to smooth the analytic envelope but robustly restore its unique form

by using a method for shape reconstruction from a point cloud. For this purpose we address

modern techniques for processing spatial images and multidimensional data. These methods rely

on Euclidean geometry and have been proven valid in a number of substantive areas (see, e.g.

(Abbott & Tsay 2000)). In particular, they become very popular in geophysics these days. Most

applications of such techniques, however, are related to geophysical inversion (for example,

searching the parameter space that contains models of acceptable data fit – neighborhood

algorithm). On the contrary, applications to time series data haven’t been widespread thus far,

possibly due to difficulties with constructing relevant metrics and similarity measures.

In this article we invoke k-order α-shape – a new computational-geometry tool for shape

restoration from a point cloud. This enables us to reconstruct the coda shape and do without

smoothing. We show that application of k-order α-shape to seismic envelopes leads to robust

delineation of the coda decay curve and yields consistent derivative estimates. On a broader

scale, the proposed algorithm gives a practical means for shape recovery from any signal-plus-

noise dataset given as time series.

2 The tool: k-order α-shape α-shape (Edelsbrunner et al. 1983) is a classical computational-geometry tool for restoring the

shape out of a point cloud. Depending on application area, the input point cloud can be obtained

in various ways. One common way is sampling a physical object or an environmental

phenomenon.

The definition of α-shape of a point set is as follows:

Definition [α-shape of a point set P]. Let P R2 be a set of points in the plane. The α-shape of P

connects all pairs of points p,q P for which there exists a disk of radius α having p,q on the

boundary and having no points of P inside. For example, with α = 0 no pairs of points get

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connected in the α-shape. In the other extreme case, when α ∞, the α-shape is the convex hull

of P.

α-shape formally captures the intuitive notion of “shape” of a point set (Fig. 1, left). The value of

α controls the level of details with which the shape is restored. Over the years, α-shapes were

used in numerous domains (see e.g. Edelsbrunner & Mucke 1994; Teichmann & Capps 1998;

Albou et al. 2009; Zhou W & Yan H. 2012). Many useful properties of α-shapes have been

established theoretically and confirmed in practice in diverse applications ranging from graphics

to biology.

2.1 k-order α-shape: handling outliers

Real-world data are often noisy and contain outliers that distort the α-shape (Fig. 1, middle). To

address this issue, Krasnoshchekov & Polishchuk (2014) have recently introduced k-order α-

shape – an extension of α-shape, capable of ignoring a certain amount of outliers and providing a

more robust shape reconstruction. Figure 1(left) shows α-shape of a point set, nicely

reconstructing the cloud shape. In Fig. 1(middle) two outliers are added. Now the α-shape

appears to miss an important feature of the set – the inner hole. The idea of k-order α-shape was

prompted by trying to cope with outliers by allowing few points to reside inside the α-disks

defining the shape. In Fig 1(right) 1-order α-shape of the set with the outliers is shown. It can be

seen how k-order α-shape with k > 0, being less sensitive to the outliers, provides a more robust

result.

Formally, k-order α-shapes are defined similarly to α-shapes using disks of radius α. Unlike with

the standard α-shapes defined by empty disks, the disks that define k-order α-shapes contain k

points of P. Specifically, a disk of radius α is called k-full if exactly k points of P reside inside the

disk. The points of P, contained in k-full disks are called outside; the points that are not outside

are called inside.

Definition [k-order α-shape of a point set P]. k-order α-shape is the α-shape of the inside points.

With this definition the α-shape is the k-order α-shape with k = 0. Algorithmic and combinatorial

properties of k-order α-shapes are given in (Krasnoshchekov & Polishchuk 2014). There we show

that k-order α-shapes are connected to the k-order Voronoi diagram in the same way in which α-

shapes are connected to the Voronoi diagram. Our video demonstrating properties of k-order α-

shapes “in action” can be viewed from (Krasnoshchekov et al. 2010). By playing with the

interactive web-applet at http://www.cs.helsinki.fi/group/compgeom/kapplet/ one can get a feel

of how k-order α-shapes work.

2.2 Reconstructing inner shape

The α-shape was originally designed to delineate the outer shape of the point set; following the

outer boundary is enabled by using α-disks empty of points of P. In particular, extreme points in

the data may significantly influence the α-shape. With positive k, k-order α-shape locally “shaves

off” k extreme points of P – specifically, the points that are outside. As a result, the inner shape

of the point set is obtained. See Figure 2 for an example.

We remark that α-shapes (and k-order α-shapes) are non-parametric techniques. Indeed, no a

priori assumption is made about the reconstructed shape. That is, it is not assumed that the shape

is a graph of a linear, exponential, logarithmic, or any other function. In contrast, in parametric

regression the general form of functional dependence between the input and output is assumed to

be known in advance – it is postulated that the dependence belongs to some known family of

functions. The regression task is then to find the best values for the parameters that describe the

function (i.e., the search is performed in the parameters space).

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3 Algorithm and processing The previous section described advantages of using k-order α-shape for shape reconstruction

purposes. Here we present an algorithm for analyzing time series data with k-order α-shapes. We

report on an application of our algorithm to processing seismic envelopes. Our main purpose is to

reconstruct a unique smooth curve representing the continuous shape of seismic coda from the

shape’s discrete representation given in the form of seismic envelope. That is, in terms of spatial

reconstruction we view this discrete dataset as a result of sampling of the original coda shape.

Basing on fundamental assumptions in body wave propagation, we also assume that the sampling

is sufficient to enable reconstruction of the desired shape with reasonable error. In contrast to

moving averaging that smooths envelope by averaging constant number of nearby points, the

presented algorithm allows one to infer spatial relationships among the dataset points. And the

whole reconstruction procedure thus consists of finding spatial relationships between points of an

unorganized dataset and estimating the relevant local element of the target shape.

The technical task is thus to restore the sought shape out of the point cloud that also includes

noise. Our approach treats seismic noise separately from the target coda shape. k-order α-shape

has wide variety of capabilities including the above-demonstrated options of strict rejection of

extreme values. Figuring on these capabilities, we employ k-order α-shape to robustly extract

seismic coda shape from analytic envelope serving as the input dataset to the shape

reconstruction procedure. The dataset is essentially time series with the following features. First,

the dataset includes a passage with seismic noise, where the sought shape is knowingly absent.

Second, it has a passage with assumed presence of the target shape complicated by presence of

extreme points and seismic noise. We remark that straightforward application of the classical α-

shape is evidently not effective because the resulting shape would be seriously distorted by

extreme points and characterize basically the outer shape of the point cloud, while we are

intuitively searching for the “inner” one. This prompted us to invoke k-order α-shape – the

generalization of α-shape, capable of handling extreme values and reconstructing the inner shape

of a dataset.

3.1 Processing of time series The algorithm: We apply the k-order α-shape to time series analysis as follows. For every time

tick we have two disks of radius α – the upper disk and the lower disk. The upper disk is put

above the data points and is moved down; the disk is stopped as soon as it contains k data points

(Fig. 3, left). Similarly, the lower disk is put below the data and is moved up until gathering k

points inside. Our output at the time tick is the midpoint of the segment that connects the stopped

disks’ centers.

To build up intuition about the algorithm’s output, consider the extreme cases of α = 0 and α = ∞.

In the former case, the output will be just the original time series; in the latter case the output will

be the straight line at the average level between the kth

max and kth

min of the time series.

Therefore, the algorithm should be run with reasonable choice of α and k. A practical method for

choosing α and k for analytic envelope of seismic trace is presented in Appendix.

The proposed algorithm is applicable and yields an output shape on time series of any nature. We

remark that seismograms are particularly amenable to the algorithm due to the presence of

seismic noise. This unique feature of such data allows us to adjust α and k separately for each

trace. The selection is based on a general presumption that output from the shape reconstruction

procedure applied to seismic noise must be close to the straight line.

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Similarly to standard time series processing methods, the output of k-order α-shape at a time tick

is influenced only by values at time ticks that are close (within plus minus α) to the time tick.

However, traditional time series methods treat the local data in a predefined manner, e.g., by

weighting the points so that further points have smaller weights. k-order α-shape, on the contrary,

has no predefined weighting scheme and produces the output at every time tick based on

interaction of the α-disks with the actual spatial distribution of the local data. k-order α-shape

method is robust: small variations in the location of input points do not influence the resulting

shape (Fig. 3, right). Indeed, at each time tick the output of our algorithm is defined by the points

that made the upper and the lower disks stop. Variations in positions of the other points (both

inside and outside the disks) are not crucial for the output at the time tick.

The overall algorithmic flow of the processing is presented in Fig. 4. Its individual steps are

discussed above. We made available our MATLAB code for the above algorithm at

http://www.cs.helsinki.fi/group/compgeom/seism.zip. The code reads local bulletin, loads raw

data files in mseed format, computes envelopes of records for the specified time intervals, runs

the k-order α-shape algorithm and outputs the restored coda shape of the local event. All the

procedures are fully automated; the user can also adjust the code parameters to suit her particular

application. More detailed instructions are available in the code archive or by contacting the

authors. Finally, an example for visual comparison of averaged and α-shaped envelopes is given

in Online Supplementary Materials (Fig. S1).

3.2 Application to synthetic data We tested the performance of k-order α-shape on a synthetic problem instance in which the true

shape is known in advance and can be compared with the output of the algorithm. We designed

the dataset to resemble real seismic data. The dataset consisted of two parts: the straight line at a

constant amplitude level followed by the 3-D solution of time-dependent Boltzmann equation

taken as a realistic example of synthetic coda envelope (Paasschens 1997). The latter part is the

true shape that k-order α-shape has to reconstruct. It essentially consists of the ballistic peak

followed by a tail due to waves which have undergone a single forward scattering event, and the

diffusion-type tail (see e.g. (Hoshiba 1995) or (Calvet & Margerin 2013) for illustrations of the

coda shape just after the direct wave). We then superposed Rayleigh-distributed fluctuations

(Nakahara & Carcolé 2010; Anache-Ménier et al. 2009) all through the dataset. The created

synthetic trace was 45 min long at sampling frequency of 100 Hz giving a total of 270000 data

points (just like real data we will consider below).

By way of proof of concept, we applied the above k-order α-shape algorithm to the created

synthetic trace and obtained the output curve. The results are presented in Fig. 5. It can be seen

the output curve almost coincides with the true shape without prominent shifts or excursions all

through the preceding noise, single scattering region and the diffusion tail (Fig. 5, left). The only

shape detail somehow distorted by the processing is the ballistic peak due to direct arrival – we

observe a sort of “shoulder” instead (Fig. 5, right). To quantify the difference between the output

curve and the true shape we calculated the root-mean-square error of the reconstruction, which

made few tenths of percent.

3.3 Real data trial More testing was performed then on real data. It was necessary because the above generated

synthetic dataset was missing some important features of the real seismic noise. For example, it

does not take into account any temporal correlation in the noise inherently present in any seismic

data, or low frequency variations in the mean noise level. To evaluate our algorithm on real data

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we processed a broadband record of local Fennoscandian event. The crust earthquake with ML =

2.4 was registered 109 km away by temporal station LP23 of LAPNET network. Prior to

application of the algorithm, we increased signal to noise ratio by frequency filtering between 1

and 15 Hz, evaluated envelope of the filtered trace and took logarithm of the envelope. Figure 6

presents the shape reconstruction output from the trace. In contrast to wild oscillations of the

input envelope, the output curve is a function smoothly decaying from the first arrival peak to the

level of seismic noise and drawing low frequency waves around this level before and after the

event.

3.4 Error analysis To evaluate standard error and stability of the output curve we used a bootstrap-type resampling

algorithm (Efron & Tibshirani 1991). For this purpose we first created multiple (N) realizations

of the original envelope. In each realization we randomly removed 25 per cent of data points

while keeping synchronization of the bootstrapped pseudo-envelope. The latter means that some

time ticks become “empty”, i.e. have no corresponding envelope value. Such lacunas

momentarily and locally decrease the original sampling rate and make the re-sampled

bootstrapped dataset unevenly spaced.

Each of the N created pseudo-envelopes was processed with the above algorithm, which yielded

the population of N output curves. As a compromise between statistical significance and

computational efficiency we used N = 50; using a somewhat different value of N does not

significantly affect our results. Figure 6 plots overlaid the whole population of 50 bootstrapped

solutions and the unperturbed output curve. Visually, all through the curve the data points of the

processed pseudo-envelopes are evenly distributed around the original output curve and make no

local clusters; the unperturbed curve doesn’t systematically deviate from the assumed imaginary

center of the “tube” formed by the population.

To quantify error, at each time tick we calculated the relative standard deviation of the whole

population of output datapoints. Except for the time period corresponding to arrival of primary

waves, the error was well below 1% (Fig. 7). In Figure 8 we present distributions of error below

and above 5%. The total number of time ticks where the error was above 5% made 767, which

was just 0.5% of the total length of the trace. All these points occurred in the region of the sharp

growth of envelope amplitude associated with first arrival of P. Restoring coda shape in this

region of abrupt changes is beyond the scope of this paper and irrelevant for coda studies.

The above considerations justify the use of k-order α-shape for extraction of coda shape from

seismograms. In the following section we report on the results of application of k-order α-shape

reconstruction tool to determining one of important coda characteristics - the coda duration.

4 Coda duration measurement: a concrete application In this section we report on an application of k-order α-shape algorithm to measurement of total

duration of seismic coda after a local event. The total coda duration in seconds is usually

measured between the P-wave arrival and the time when S-coda amplitudes decrease to the level

of microseisms (Bath 1981). The relation between logarithm of coda duration, epicentral distance

and local magnitude is important in studies of local seismicity and is usually derived for each

region individually (Soloviev 1965). Here we measure coda durations of five local events

occurred in Fennoscandia and compare the outputs with previously obtained results for this

region.

The list of analyzed events with relevant source parameters from Institute of Seismology of the

University of Helsinki (www.seismo.helsinki.fi/english/bulletins) is given in Table 1. Duration of

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seismic coda after a local earthquake was measured on vertical records of broadband instruments

of permanent and temporal Finnish stations installed under auspices of LAPNET/POLENET

project. Seismic stations were equipped with the same unified instruments, which eliminated

inconsistency due to equipment diversity. The epicentral distance to seismic stations varied from

5 to 516 km. Each event was registered by at least 10 nearby stations, and all traces were

processed in exactly the same way independently of epicentral distance or magnitude.

Each record was processed as follows. To increase signal to noise ratio we bandpass filtered each

vertical trace between 1 and 15 Hz. Such frequency filtering enabled to visualize the picture of

coda decay. Then we hand-pick the onset of the first arrival of P (alternatively, the onsets can be

taken from the source bulletin). Each onset was used as a start time from which the event coda

duration was measured on the record. We then calculated analytic envelopes and applied the k-

order α-shape algorithm to measure the total duration of seismic coda. The analyzed time period

of traces spanned 15 minutes prior to first arrival of P and half an hour past it, which is sure to

include the whole coda of any local event.

Our dataset includes records of weaker events whose seismic coda length is notoriously hard to

measure. Indeed, the analyzed events with ML < 2.0 from the above list were almost

unobservable by eye on raw records of more distant stations, so application of moving averaging

to such small events may virtually smooth out the weak signal that barely stands above the

background seismic noise even after frequency filtering.

We identified the coda end as the time instant when the output curve first drops below the mean

level of the preceding background noise (although background seismic noise is not ideal and

reveals manifold biases and anomalies, its mean level over the time period immediately

preceding the event is frequently used as a natural benchmark). That is, we used the simplest

definition of coda duration as the time elapsed from first arrival of primary waves until the coda

falls beneath some absolute amplitude level. In general such approach is not easy to implement,

in particular due to rapid oscillations of the analyzed time series (either the original seismogram

or its envelope). To overcome it, there have been presented a number of tricks involving signal-

to-noise amplitude ratios, relative amplitude measurements, etc. In contrast, our output curve

representing the coda shape is a smoothly decaying almost non-oscillating function that supports

uncomplicated time measurements on the base of simple criteria.

An example of output curves along with measured coda durations are shown in Fig. 9. The 14

restored shapes of seismic codas recorded at epicentral distances between 5 and 308 km from a

local event (#1 in Table 1) are displayed overlaid and aligned onto the origin time. The presented

shapes are consistent with theoretical predictions as to constant decay gradient and the total time

required to dump seismic energy radiated from a source. According to Fig. 9, variability in this

latter parameter estimated by our automatic procedure is roughly about 25 seconds, which is not

large for a weak event.

To compare total coda durations with previous results we turned to the notion of duration

magnitude. The linear relationship between logarithm of seismic coda length τ and epicentral

distance Δ in km to the local event with magnitude M is most commonly given by

M = a*Δ + b*log10τ + c (1)

or

M = a*Δ + b*log2

10τ + c (2)

where a, b and c are constants. The values of a, b and c for a given station or region are usually

estimated using the regression analysis of extensive datasets of τ versus Δ and are subject to

significant changes from one region to another. For a given magnitude the linear relationship

between logarithm of τ and Δ is defined by the ratio a/b. This ratio for formula (1) was estimated

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as -1.2x10-3

for Japan (Tsumura 1967) and -3.4x10-4

for Northern California (Hirshorn et al.

1987). The Fennoscandian magnitude scale on the base of coda duration was derived by

Wahlstrom (1980) who used formula (2). The Fennoscandian linear relationship coefficient a/b in

the formula was estimated as -2.9x10-3

, which is not far from the analogous scale for Central

California yielding -4.4x10-3

(Bakun 1984).

In order to estimate output quality of our automatic coda duration measurements, for each event

from our dataset we plotted squared logarithm of total coda length in seconds versus epicentral

distance in km. The results for two events with the smallest and the largest magnitudes (M = 1.4

and M = 2.4, respectively) are given in Fig. 10. The plots clearly display the expected dependence

of coda duration decrease with epicentral distance. The last column of Table 1 shows the relevant

linear regression estimates for all 5 events. The presented values of linear relationship coefficient

a/b exhibit some variation, but the average value made (-3.6 ± 1.0)x10-3

, which is very close to

the previous estimate by Wahlstrom (1980).

5 Conclusion and future work In this paper we presented a new algorithm for restoring seismic coda shape from analytic

envelopes. We successfully applied the algorithm to measuring total coda length on local records

of weak events. The relation coefficient between coda duration and epicentral distance is

consistent with the earlier findings for the region.

The new algorithm is based on α-shape - a well established tool for shape reconstruction, that

previously has been successfully applied in various domains. We demonstrate how the new tool

provides robust non-parametric estimation of shape of a point cloud given as time series. A useful

feature of our algorithm is that it is not oriented towards any specific absolute criterion of output

quality, so in any particular application the user is free to choose the most suitable quality

measure.

We envision that k-order α-shapes will be applied to other seismological tasks such as duration

and local magnitude estimations, studies of seismic attenuation derived from coda decay and fine

temporal variations in coda shape. It would also be interesting to see if k-order α-shape is

applicable to other geophysical datasets (e.g. geomagnetic).

Appendix To choose α and k for each trace we use the part of the record that contains background noise

only (Fig. 4). We set α equal the standard deviation of the noise. To choose k we take 1000

random time ticks and at each tick place two α-disks – one with the center 2α above the mean

level of noise, and the other with the center 2α below the mean level; k is taken as the average

number of data points over the 2000 disks. We then apply the k-order α-shape algorithm to the

noise, and measure the similarity between the output shape and the mean level using root-mean-

square deviation (RMSD). Intuitively, the α-disks are expected to “shave off” the noise so that a

straight line is obtained in the output. If the variability of the output curve is too high, we scale

the time axis and recompute k with the same procedure (averaging over the 2000 disks at random

time ticks). When larger scaling is applied to the time axis, the chosen value for k also becomes

larger, which implies a smaller-variability output curve (decreased RMSD). One stops at the

smallest scale for which the RMSD gets below the pre-defined level. RMSD between 0.05α and

0.005α usually suffices. Using higher scales would lead to an oversmoothed output.

Overall, the described method follows the standard paradigm in artificial intelligence and

machine learning: splitting data into training and validation sets. We emphasize that α and k are

chosen separately for each trace, i.e. these operational parameters are adjusted individually for

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each record in question. Furthermore, for any particular record, the choice of α is independent of

the time axis scaling, while k is chosen separately for each scale. We also remark that RMSD is

not the only possible measure of quality of noise fit; any other goodness measure can be used just

as well.

Acknowledgements

The authors wish to thank LAPNET/POLENET Workgroup headed by Elena Kozlovskaya from

Sodankylä Geophysical Observatory for providing high-quality broadband seismic records of

LAPNET seismic array. We also thank the anonymous reviewers for their helpful input and

researchers at Helsinki Institute of Seismology for discussions. This work was supported by the

Academy of Finland grants 1138520 and 261019 and by Research Funds of University of

Helsinki.

The POLENET/LAPNET project was a part of the International Polar Year 2007-2009 and a part

of the POLENET IPY consortium. Organisations participated in the POLENET/LAPNET project

are: (1) Sodankylä Geophysival Observatory of the University of Oulu(Finland), (2) Institute of

Seismology of the University of Helsinki (Finland), (3) University of Grenoble (France), (4)

University of Strasbourg (France), (5) Institute of Geodesy and Geophysics, Vienna University of

Technology (Austria)j, (6) Geophysical Institute of the Czech Academy of Sciences, Prague

(Czech republic), (7) Institute of Geophysics ETH Zürich (Switzerland), (8) Institute of

Geospheres Dynamics of the Russian Academy of Sciences, Moscow (Russia), (9) The Kola

Regional Seismological Centre, of the Russian Academy of Sciences (Russia), (10) Geophysical

Centre of the Russian Academy of Sciences, Schmidt Institute of Physics of the Earth of the

Russian Academy of Sciences (Russia), (11) Swedish National Seismological Network,

University of Uppsala (Sweden), (12) Institute of Solid Earth Physics, University of Bergen

(Norway), (13)NORSAR (Norway), (14) University of Leeds (UK). Equipment for the

temporary deployment was provided by RESIFSISMOB, FOSFORE, EOST-IPG Strasbourg

Equipe sismologie (France), seismic pool (MOBNET) of the Geophysical Institute of the Czech

Academy of Sciences (Czech republic), instrument pool of Sodankylä Geophysical Observatory

(Finland), Institute of Geosphere Dynamics of RAS (Russia), Institute of Geophysics ETH

Zürich (Switzerland), Institute of Geodesy and Geophysics, Vienna University of Technology

(Austria), University of Leeds (UK). Funding agencies provided support for organisation of the

experiment are: Finland : The Academy of Finland (grant No. 122762) and University of Oulu;

France: BEGDY program of the Agence Nationale de la Recherche, Institut Paul Emil Victor and

ILP (International Lithosphere Program). task force VIII.; Czech Republic: grant No.

IAA300120709 of the Grant Agency of the Czech Academy of Sciences; Russian Federation :

Russian Academy of Sciences (programs No 5 and No 9). POLENET/LAPNET Working Group:

Elena Kozlovskaya (1), Helle Pedersen (3), Jaroslava Plomerova (6), Ulrich Achauer (4), Eduard

Kissling (7), Irina Sanina (8), Teppo Jämsen (1), Hanna Silvennoinen (1), Catherine Pequegnat

(3), Riitta Hurskainen (1), Robert Guiguet (3), Helmut Hausmann (5), Petr Jedlicka (6), Igor

Aleshin (10), Ekaterina Bourova (3), Reynir Bodvarsson (11), Evald Brückl (5), Tuna Eken (6),

Pekka Heikkinen (2) Gregory Houseman (14), Helge Johnsen (12), Elena Kremenetskaya (9),

Kari Komminaho (2), Helena Munzarova (6) Roland Roberts (11), Bohuslav Ruzek (6), Hossein

Shomali (11), Johannes Schweitzer (13), Artem Shaumyan (8), Ludek Vecsey (6), Sergei

Volosov (8)

10

Table 1. Source parameters of analyzed events and relation coefficients (a/b) between coda

duration and epicentral distance estimated by least squares for each event.

# Origin Source Location and Magnitude # of

stations

a/b

Date

yyyy/mm/dd

Time

hh:mm:ss.0

latitude

(degrees)

longitude

(degrees)

depth

(km)

ML

1 2008/09/27 02:20:11.9 65.987 29.963 18 2.4 31 (-2.2 ± 0.4)x10-3

2 2007/11/13 03:29:07.8 66.041 29.519 17.1 1.7 13 (-5.2 ± 1.0)x10-3

3 2008/01/17 03:46:43.2 65.796 29.866 14.2 1.7 12 (-3.9 ± 1.1)x10-3

4 2008/08/10 19:02:28.9 65.592 27.866 14.8 1.6 10 (-3.9 ± 1.4)x10-3

5 2008/04/06 05:12:05.1 65.876 29.868 9.8 1.4 10 (-3.0 ± 0.6)x10-3

References Abbott, A., Tsay, A., 2000. Sequence Analysis and Optimal Matching Methods in Sociology:

Review and Prospect, Sociological Methods & Research, 29, 3–33.

Albou, L. P., Schwarz, B., Poch, O., Wurtz, J.M., Moras, D., 2009. Defining and characterizing

protein surface using alpha shapes, Proteins, 76(1), 1-12.

Aki, K., 1969. Analysis of seismic coda of local earthquakes as scattered waves, J. Geophys.

Res., 74, 615—631.

Anache-Ménier, D., van Tiggelen, B.A. & Margerin, L., 2009. Phase Statistics of Seismic Coda

Waves, Phys. Rev. Lett., 102, #24, 248501.

Bakun, W. H., 1984. Magnitudes and Moments of Duration, Bull. Seismol. Soc. Am., 74 (6),

2335–2356.

Bakun, W. H., 1984. Seismic Moments, Local Magnitudes, and Coda-Duration Magnitudes for

Earthquakes in Central California, Bull. Seismol. Soc. Am., 74, 439–458.

Bath, M., 1981. Earthquake magnitude - recent research and current trends, Earth Science Rev.,

17, 315-398.

Calvet, M. & Margerin, L., 2013. Lapse‐ Time Dependence of Coda Q: Anisotropic

Multiple‐ Scattering Models and Application to the Pyrenees, Bull. Seism. Soc. Am., 103 (3),

1993-2010.

Edelsbrunner, H., Kirkpatrick, D. G. & Seidel, R., 1983. On the shape of a set of points in the

plane, IEEE Trans. Inform. Theory, 29, 551–559.

Edelsbrunner, H., Mucke, E. P., 1994. Three-dimensional alpha shapes, ACM Trans. Graph.

13(1), 43—72.

Efron, B., Tibshirani, R., 1991. Statistical data analysis in the computer age, Science, 253, 390–

395.

Hirshorn, B., Lindh, A., Allen, R., 1987. Real Time Signal Duration Magnitudes from Low-gain

Short Period Seismometers, U.S. Geol. Surv., Open-File Rep, 87–630.

Hoshiba, M., 1995. Estimation of nonisotropic scattering in western Japan using coda wave

envelopes: Application of a multiple nonisotropic scattering model, J. Geoph. Res., 100, B1, 645-

657.

Krasnoshchekov, D., Polishchuk, V., 2014. Order-k α-hulls and α-shapes, Information

11

Processing Letters, 114, 76—83.

Krasnoshchekov, D., Polishchuk, V., Vihavainen, A., 2010. Shape approximation using k-order

alpha-hulls, Symposium on Computational Geometry, 109-110, http://youtu.be/KeRKgv4Slts

Nakahara, H., Carcolé, E., 2010. Maximum-Likelihood Method for Estimating Coda Q and the

Nakagami-m Parameter, Bull. Seismol. Soc. Am., 100 (6), 3174–3182, doi: 10.1785/0120100030.

Paasschens, J.C.J., 1997. Solution of the time-dependent Boltzmann equation, Phys. Rev., E56,

1135–1141.

Rautian, T.G., Khalturin, V.G., 1978. The use of the coda for the determination of the earthquake

source spectrum, Bull. Seismol. Soc. Am., 68, 923–948.

Sato, H., Fehler, M.C., 1998. Seismic wave propagation and Scattering in the homogeneous

Earth, AIP Press, New York.

Soloviev, S.L., 1965. Seismicity of Sakhalin, Bull. Earthq. Res. Inst., 43, 95—102.

Teichmann, M., Capps, M. V., 1998. Surface reconstruction with anisotropic density-scaled alpha

shapes, in IEEE Visualization, 67—72.

Tsumura, K., 1967. Determination of Earthquake Magnitude from Total Duration of Oscillation,

Bull. Earthquake Res. Inst., Tokyo Univ., 15, 7–18.

Wahlström, R., 1980. Duration Magnitudes for Swedish Earthquakes, Geophysica, 16, #2, 171-

183.

Zhou.W., Yan, H., 2012 Alpha shape and Delaunay triangulation in studies of protein-related

interactions, Briefings in Bioinformatics, doi: 10.1093/bib/bbs077.

Figure 1. Left: A point set and its α-shape. Middle: Few outliers (shown with asterisks) distort the

restored shape. Right: k-order α-shape (k = 1) of the set with outliers brings the shape back.

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Figure 2. The k-order α-shape (k = 15) effectively restores the inner shape of the data cloud.

Figure 3. Application of k-order α-shape to time series. Left: The upper (resp. lower) disk is

pushed down (resp. up) from above (resp. below) the data; the disk stops when there is exactly k

= 4 points inside it. The output at the time tick is the average of the disks’ centers. Right: some

points (both inside and outside the disks) have changed their locations slightly; nevertheless. the

output at the time tick does not change.

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Figure 4. The overall algorithmic flow: Seismic record is split into the noise and coda. The noise

is used to choose α and k (e.g. with the help of RMSD – see Appendix). The chosen α and k are

applied to coda to restore its shape. The dashed line is the mean level of background noise (also

shown are four disks at random time ticks); the output of the algorithm is in black.

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Figure 5. Application of k-order α-shape to synthetic example: the result of processing the whole

45-min long synthetic dataset (left), and the 5-min excerpt corresponding to the passage that

represents the first arrival of P (right). In both panes the true shape (black) and the restored shape

(white) are overlaid on the synthetic dataset (grey) with Rayleigh distributed noise.

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Figure 6. Error analysis: the unperturbed output (black) and 50 bootstrapped solutions (white)

overlaid on the original analytic envelope (grey).

16

Figure 7. Quantifying the error: synchronized plot of bootstrapped population of output curves

(above) and relative standard deviation (below) versus time ticks. The range of higher error

correlates with first arrival of P-waves and comes as a “shoulder” on the output curve.

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Figure 8. Histogram of error below (left) and above (right) 5%.

Fig. 9. Results of coda shape restoration procedure applied to 14 envelopes of a local event (see

event #1 in Table 1 for seismic source parameters).

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Fig. 10. Linear regression results for events #5 (left) and #1 (right) from Table 1.